the more you know about weed and its preferred habitat and dissemination will help. Unless it has some unique characteristic that can be identified remotely, alot will depend on what you can gather from botanists and ecologists. If the seeds move with wind, water, birds or animals, more difficulty. If the plant is specific to conditions such as cultivated or hot burns on uplands, normally within vicinity of roads, those might be characteristics you can conduct GIS analysis on and stratify a survey. If you can develop a remote sensing signature for the weed, you may be able to detect with high resolution or standard aerial photography. Also there is increased use with the toy like small aircraft with movie cameras to assess conditions from closer to ground level. There was a researchgate question on their use for monitoring agriculture field conditions a few months back.
You may want to give CIAT's FloraMap a try (it is free for download). It can produce probability plots on the possible occurrence of plant species based on climatic data. You need to have some data on the actual distribution of that plant from herbaria or germ plasm collections, or you can use the GBIF database to obtain such data to start with (I think you have to register with GBIF but you can use it for free). Basically, FLoraMap extracts the climatic data from these collection points and searches for locations with similar climates. The probability plots produced by FloraMap are shp-files which can be further processed in any other GIS application.