Mona can use her Burst to apply the Omen debuff on enemies and allow teammates to deal more damage. This is most commonly seen in Freeze teams. Mona's main role on a Freeze team is to provide Omen debuff from her Burst as it is extendable via Freeze mechanics.
Note that you must apply Cryo before casting Mona's Burst for the extension to happen. Omen Freeze extension can be done by reapplying Cryo and Hydro aura consistently (this can be done with Cryo / Hydro infused Venti Burst + on-field Cryo / Hydro aura).
Mona's Burst applies bubble and Omen debuff upon cast, the count down timer only begins when the bubble is broken.
Frozen enemies can have their bubbles preserved as long as enemies stay in a frozen state. This technique makes Mona a priority Support for Cryo DPS teams.
Regarding Weapon Choices:
Thrilling Tales of Dragon Slayers: Generally the best option on Support Mona because she can furthermore buff the ATK of your Main DPS. This however comes with an assumption that with this weapon, you still have 100% Burst uptime. If you do not, pick the other Energy Recharge weapon options for more Burst uptime.
A Thousand Floating Dreams: If the team does not need Elemental Mastery this weapon should not be used on Mona. If the team does need Elemental Mastery and does not benefit heavily from ATK%, this weapon can compete with .
Regarding Artifact Choices:
Offensive stats aren't actually needed on a Freeze Support Mona, focus on Artifact Set Bonus & stack Energy Recharge for comfortabl'Burst uptime