update: It looks like I missed the mark here; I've answered about a radio identifier used for communications, whereas the question asks about a "official call sign" or designator that is used when people talk to each other about the helicopter.
tl;dr: It will likely have a ZigBee identifier between "1" and "240" unless they've turned that off somehow. DSN can't hear 900 MHz (I don't think) but Green Bank could indeed pick up "proof of life" if so inclined! FAST could as well, as could several large radio astronomy arrays like VLA and ALMA.
Addressing is also part of the application layer. A network node consists of an 802.15.4-conformant radio transceiver and one or more device descriptions (basically collections of attributes which can be polled or set, or which can be monitored through events). The transceiver is the base for addressing, and devices within a node are specified by an endpoint identifier in the range 1-240.
But unless there's some software defined radio around that can access 900 MHz, I think that only Perseverance rover will likely be used to listen to Ingenuity regularly.
However plenty of giant dishes can tune in to 900 MHz and try!
From Is there anything that could speak 900 MHz ZigBee that could hear Ingenuity helicopter besides the Perseverance rover?
Wikipedia' Ingenuity (helicopter) links to Mars Helicopter Technology Demonstrator (Canham et al. 2018, AIAA Atmospheric Flight Mechanics Conference, also here and archived) which says:
F. Telecommunication System
Once separated from the host spacecraft (lander or rover), the Mars Helicopter can only communicate to or be commanded from Earth via radio link. This link is implemented using a COTS 802.15.4 (Zig-Bee) standard 900 MHz chipset, SiFlex 02, originally manufactured by LS Research. Two identical SiFlex parts are used, one of which is an integral part of a base station mounted on the host spacecraft, the other being included in the helicopter electronics.
These radios are mounted on identical, custom PC boards which provide mechanical support, power, heat distribution, and other necessary infrastructure. The boards on each side of the link are connected to their respective custom antennas. The helicopter antenna is a loaded quarter wave monopole positioned near the center of the solar panel (which also serves as ground plane) at the top of the entire helicopter assembly and is fed through a miniature coaxial cable routed through the mast to the electronics below. The radio is configured and exchanges data with the helicopter and base station system computers via UART.
One challenge in using off-the-shelf assemblies for electronics systems to be used on Mars is the low temperatures expected on the surface. At night, the antenna and cable assemblies will see temperatures as low as −140 °C. Electronics assemblies on both base station and helicopter will be kept “warm” (not below −15 °C) by heaters as required. Another challenge is antenna placement and accommodation on the larger host spacecraft. Each radio emits approximately 0.75 W power at 900 MHz with the board consuming up to 3 W supply power when transmitting and approximately 0.15 W while receiving. The link is designed to relay data at over-the-air rates of 20 kbps or 250 kbps over distances of up to 1000 m.
A one-way data transmission mode is used to recover data from the helicopter in real time during its brief sorties. When landed, a secure two-way mode is used. Due to protocol overhead and channel management, a maximum return throughput in flight of 200 kbps is expected while two-way throughputs as low as 10 kbps are supported if required by marginal, landed circumstances.
From here:
Link Budget
From this answer which is from this answer:
$$ P_{RX} = P_{TX} + G_{TX} - L_{FS} + G_{RX} $$
- $P_{RX}$: received power by spacecraft
- $P_{TX}$: transmitted power by wristwatch
- $G_{TX}$: Gain of wristwatch's transmitting antenna (compared to isotropic)
- $L_{FS}$: Free space Loss, what we usually call $1/r^2$
- $G_{RX}$: Gain of spacecraft's receiving antenna (compared to isotropic)
$$G \sim 20 \times \log_{10}\left( \frac{\pi d}{\lambda} \right)$$
$$L_{FS} = 20 \times \log_{10}\left( 4 \pi \frac{R}{\lambda} \right).$$
At 900 MHz (33 cm) the Green Bank Telescope 100 m dish has a gain $G_{RX}$ of 56 dBi at best. I don't think the DSN receivers operate at 900 MHz, but that's prime territory for radio astronomers.
Free space path loss at 1 AU is then 255 dB. Mars varies from 0.3 to 3 AU so squaring that we'll add a +/- 10 dB at the end.
We'll just call the ZigBee's antenna "isotropic" so the antenna gain is 0 dBi by definition. The transmit power is of order 1 Watt or 0dBW.
We have then a received power of -199 +/- 10 dBW. That's $10^{-19}$ to $10^{-21}$ watts. Assuming Green Bank has a 100 Kelvin cooled front end the thermal noise $k_B T$ is about $10^{-21}$ watt seconds, so all things being ideal it should be barely able to not only unambiguously identify the signal and confirm its Doppler shift but also detect variations in the range of 1 to 100 Hz depending on distance.
That's too slow to interpret data directly if it's at high speed for the Rover, but it might allow for multi-path interference detection as the helicopter flies over terrain.
I don't know what the lowest possible data rate is that Ingenuity can transmit, and I don't know if it will produce a ZigBee beacon, but if one were very, very ambitious and/or imaginative one could think about listening for that; you'd digitize a long stretch then apply a clever algorithm to find it in post-processing.